Every computable universal algebra has a finitely presented expansion; on the other hand, there are examples of finitely generated, computably enumerable universal algebras with no finitely presented expansions. It is natural to ask whether such examples can be found in well-known classes of algebras such as groups and semigroups. Here we build an example of a finitely generated, infinite, computably enumerable semigroup with no finitely presented expansions. We also discuss other interesting computability-theoretic properties of this semigroup.
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The paper is based on the invited talk given by B. Khoussainov at the Mal’tsev meeting 2011 dedicated to the 60th birthday of S. S. Goncharov.
(D. R. Hirschfeldt) Supported by the National Science Foundation of the United States, grants DMS-0801033 and DMS-1101458.
(B. Khoussainov) Supported by the Marsden Fund of the Royal Society of New Zealand.
Translated from Algebra i Logika, Vol. 51, No. 5, pp. 652–667, September-October, 2012.
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Hirschfeldt, D.R., Khoussainov, B. Finitely presented expansions of computably enumerable semigroups. Algebra Logic 51, 435–444 (2012). https://doi.org/10.1007/s10469-012-9203-8
- computably enumerable semigroup
- finitely presented expansion